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Educatioal  Purification  No.  124.         Div.  of  Elementary  Instruction  No.  27 


DIAGNOSTIC  STUDY 

IN 

ARITHMETIC 

BY 

MARGARET  HAYES 
Rural  School  Supervisor,  Craven  County 


Published  by  the 

State  Superintendent  op  Public  Instruction 

Raleigh,  N.  C. 


INTRODUCTION 


The  materials  in  this  bulletin  were  prepared  by  Miss  Margaret  Hayes, 
rural  supervisor  in  Craven  County.  The  purpose  of  the  investigation  was 
to  determine  as  far  as  possible  the  various  causes  for  retardation  in  the 
fundamental  processes  of  arithmetic  among  the  public  school  children 
of  that  county  and  to  suggest  remedial  measures. 

Upon  the  recommendation  of  Mr.  L.  C.  Brogden,  Director  of  the  Divis- 
ion of  Elementary  Instruction,  and  of  Miss  Hattie  Parrott  of  the  same 
division  this  bulletin  is  printed  in  the  belief  that  it  will  prove  valuable 
to  superintendents,  principals  and  supervisors  in  their  efforts  to  improve 
class  room  instruction  in  arithmetic  in  the  public  schools  of  the  State. 


state  Superintendent  of  Public  Instruction. 


9-20-28 — IM 


A  DIAGNOSTIC  STUDY  IN  ARITHMETIC 


The  need  for  this  study  arose  when  the  school  children  of  Craven 
County,  North  Carolina,  showed  persistent  retardation  in  arithmetic  fun- 
damentals. This  condition  did  not  yield  to  continued  efforts  on  the  part 
of  teachers  and  supervisor.  Therefore  a  systematic  investigation  was 
made  with  the  purpose  of  securing  data  about  this  situation,  studying 
the  bearing  on  the  problem  of  these  data,  and  arriving  at  some  helpful 
conclusions. 

Statement  of  The  Problem 

The  problem,  briefly  stated,  is  this:  What  causes  these  pupils  to  be 
unable  to  do  satisfactory  work  in  arithmetic,  and  how  can  this  condition 
be  improved? 

APPROACHES 

There  are  four  approaches  to  the  problem:  (1)  A  statistical  treat- 
ment of  the  scores  made  by  pupils  in  grades  3-8,  inclusive,  on  Woody 
Fundamentals.  (2)  A  practical  analysis  of  the  papers  just  referred  to- 
This  analysis  included  (a)  a  tabulation  of  examples  missed  in  each 
grade  and  calculation  of  percents  to  find  what  types  of  exercises  are 
generally  missed,  and  (2)  an  analysis  of  the  types  of  errors.  (3)  Indi- 
vidual diagnostic  tests  in  whole  numbers  and  fraction^  to  find  out  mental 
habits  that  retard  the  work  and  make  it  inaccurate.  The  Buswell-John 
test  was  used,  and  supplemented  by  skillful  questioning.  (4)  Close  ob- 
servation of  the  children  over  a  period  of  two  years. 

It  will  readily  be  seen  that  each  of  these  methods  of  approach  has 
certain  defects,  inherent  in  its  nature.  However,  the  strong  points  of 
one  make  up  for  the  deficiencies  of  another.  Each  method  gives  in- 
formation that  could  not  be  secured  in  any  other  way;  but  it  is  felt 
that  the  most  significant  information  was  that  gained  by  the  individual 
tests  and  the  close  observation  of  the  pupils  themselves.  These  revealed 
the   mental  habits   of  the   pupils. 

Statistical  Treatment  of  Data   From   Survey  Tests 

Scores  available  for  investigation  were  secured  by  administering  the 
Woody  Series  B  Form  1  test  in  fundamentals  to  745  pupils  in  the  Craven 
County  schools,  distributed  through  the  grades  as  follows:  195,  151, 
160,  144  and  95  pupils  in  the  fourth,  fifth,  sixth,  seventh  and  eighth 
grades,  respectively.  Tests  were  given  at  the  beginning  of  the  school 
year    1926-1927. 


4 


4  Diagnostic  Study  in  Arithmetic 

Data  are  given  in  tlie  frequency  table  below: 


(1) 

(2) 

(3) 

(4)- 

(5) 

Grade  4 

Grade  5 

Grade  6 

Grade  7 

Grade  8 

Fre- 

Fre- 

Fre- 

Fre- 

Fre- 

Score    quency 

Score    quency 

Sco7^e    quency 

Score    quency 

Score    quency 

42     1 

50      1 

56      2 

63      3 

68     1 

41     0 

48      3 

54      4 

62      3 

67     1 

40      1 

46      2 

52      6 

61      3 

66      5 

39      2 

44      5 

50     12 

60      3 

65      6 

38      2 

42     11 

48     13 

59      4 

64      3 

37      4 

40     12 

46     17 

58      7 

63      4 

36      5 

38     21 

44     23 

57     12 

62      6 

35      5 

36     25 

42     20 

56     13 

61      7 

34      3 

34     20 

40     21 

55      3 

60      6 

33      8 

32     14 

38     13 

54      9 

59      6 

32     10 

30      9 

36      8 

53      6 

58      6 

31      7 

28      9 

34      9 

52      8 

57      6 

30     12 

26      5 

32      5 

51      6 

56      6 

29      6 

24      3 

30      1 

50      3 

55      7 

28      8 

22      4 

28      2 

49      6 

54      4 

27     11 

20      4 

26      2 

48      5 

53      3 

26     19 

18      1 

24      1 

47      7 

52      6 

25     11 

16      0 

22      0 

46      5 

51      4 

24      8 

14      0 

20      0 

45      4 

50      1  ■ 

23      7 

12      1 

18      1 

44      5 

49      0 

22     10 

10      1 



43      4 

48      0 

21      7 





42      3 

47      3 

20     12 





41      7 

46      1 

19      4 





40      3 

45      1 

18      7 





39      1 

44      3 

17      3 





38      2 



16      2 





37      4 



15      6 





36      1 



14      4 





35      3 



13      3 





34      1 



12      2 

11      2 

10      1 

9      0 

8      2 

As  a  means  to  interpreting  the  data  the  following  measures  were  com- 
puted: Arithmetic  means  and  medians;  absolute  and  semi-interquartile 
ranges;  mean  and  standard  deviations;  and  total,  partial  and  multiple 
correlations  of  the  fundamental  scores  with  reading  and  reasoning  scores 
for  the  same  pupils  obtained  at  the  same  time.  When  it  was  consid- 
ered to  be   of  value   errors  of  the  measures  were   computed.  ; 

The  arithmetic  means  of  grades  4,  5,  6,  7,  and  8  are  25.98,  35.48, 
43.28,  50.98,  and  58.50,  respectively,  with  P.  E.  of  .33579,  .37289,  .34795,' 


Diagnostic  Study  in  Arithmetic  5 

.409197,  and  .39427,  respectively.  Mediums  are:  26.34,  36.36,  43.74, 
50.62    and    58.42. 

Absolute  ranges  for  grades  4,  5,  6,  7  and  8  are  34,  40,  38,  29,  and 
24  respectively;  semi-interquartile  ranges:  5.58,  6.98,  6.47,  5.41,  and 
3. 88  with  probable  errors  of  .26415,  .29334,  .27371,  .32231,  and  .31015, 
respectively.  Standard  deviations:  6.92,  6.80,  6.50,  7.28  and  5.67,  with 
probable  errors  of  .23741,  .26365,  .24363,  .289319,  and  .27876,  respect- 
ively; 10-90  percentile  ranges  are:  19.18,  15.98,  15.07,  18.45,  and  13.92, 
respectively. 

Where  reading  is  represented  by  1,  reasoning  2,  and  fundamentals 
3,  total  correlations  for  the  grades  all  taken  together  are  as  follows: 

r^3  =  .636 ;  r^.  =  .822 ;  and  r^.,  =  .715. 

(Scatter   diagram   was   used   with   Karl   Pearson's   formula.) 

Using  the  same  notation,   partial  correlations   were   found   as   follows: 

r^,3  =r.  .121;  r ^3,  .-=  .437;  r.,^^  =  .687. 

Values    found    for    multiple    correlations    are: 

R^.3  =  .72;R,^3:.=  .83;R3,,  =  .86. 

It  will  be  seen  that  fourth,  fifth,  sixth,  seventh  and  eighth  grade 
students  are  retarded  5,  7,  6,  7,  and  3  months  respectively,  using  grade 
norms.  The  lesser  retardation  of  grade  8  is  probably  due  to  the  fact 
that  no  scores  were  available  for  the  repeaters  in  this  grade.  Pupils 
in  the  rural  schools  of  this  county  are  doing  work  in  fundamentals  over 
half  a  year  before  the  national  standard.  Also,  if  a  large  number  of 
means  for  these  grades  in  the  rural  schools  of  North  Carolina  were  com- 
puted, it  is  reasonably  certain  that  the  true  means  will  lie  within  the 
Craven   County  means   +    2   P.   E.,   as   follows: 

25.30  to   26.66,  grade  ability  3.3  to   3.4. 

34.74  to  36.22,  grade  ability   4.1   to   4.2. 

42.58  to   43.98,  grade  ability  5.1   to   5.2. 

50.16  to  51.80,  grade  ability  6.1   to   6.2. 

57.72  to  59.28,  grade  ability  7.4  to  7.7. 
The  absolute  range  though  not  very  significant,  is  wide  in  all  cases, 
widest  in  grade  5  and  narrowest  in  grade  8.  The  latter  is  probably  due 
to  the  same  reason  given  for  relatively  small  retardation.  Semi-inter- 
quartile ranges  show  a  greater  clustering  of  the  scores  of  the  middle 
50  per  cent  in  grade  8,  with  grade  7  next  and  the  greatest  scattering 
in  grade  5.  10-90  percentile  ranges  give  a  closer  clustering  in  grade  4. 
Probably  there  are  more  pupils  with  low  I.  Q's,  in  grade  5,  but  the 
general  preparation  for  normal  pupils  may  be  poor  in  grade  4.  The 
most  important  measure  of  variability,  the  standard  deviation,  narrows 
the  picture  of  the  distribution  by  showing  that  68.26  per  cent  of  the 
cases  fall  between   mean   -|-   sigma,   or  in   these  cases: 

19.06  to   32.90,  grade~ability  3.1   to   3.8. 

28.68  to  46.28,  grade  ability  3.5   to   5.6. 

36.78   to  49.78,  grade  ability  4.2  to   6.1. 

43.70   to   58.26,   grade  ability   5.3   to   7.5. 

52.83   to   64.17,   grade  ability   6.4   to   8.4. 
This  shows  a  rather  wide  range  for  normal  pupils  within  the  grade. 


6  Diagnostic  Study  in  Arithmetic 

The  relationsMps  as  indicated  by  the  total  correlations  are  very  close 
for  reasoning  with  fundamentals  and  reading  with  fundamentals,  and 
close  for  reading  with  reasoning.  Evidently  the  subjects  are  all  three 
highly  interrelated  and  the  values  found  suggested  that  fundamentals 
play  the  most  important  part  in  the  trio.  Since  the  number  of  cases 
(745)  makes  this  a  fair  sample  there  is  probably  a  high  degree  of  re- 
lationship between  these  subjects  in  the  rural  schools  of  North  Caro- 
lina as  a  whole.  The  partial  correlations  give  some  insight  into  the 
nature  of  the  relationships  shown  by  the  total  correlations.  With  funda- 
mentals held  constant,  the  correlation  between  reading  and  reasoning 
reduces  from  .6.36  to  .121,  therefore  the  correlation  between  reading  and 
reasoning  is  dependent  largely  upon  the  skills  in  fundamentals.  With 
reading  held  constant  the  correlation  reduces  from  .8  22  to  .687.  Evi- 
dently some,  but  not  many  of  the  difficulties  in  problem  solving  and 
working  exercises  in  fundamentals  are  due  to  inability  to  read  the 
problems.  With  reasoning  held  constant  the  correlation  between  read- 
ing and  fundamentals  reduces  from  .715  to  .437.  This  is  an  indication 
that  the  relation  between  fundamentals  and  problem  solving  is  very 
much  stronger  than  that  between  fundamentals  and  other  subjects,  both 
when  these  other  factors  are  present,  and  when  they  are  eliminated. 
This  is  further  borne  out  by  the  fact  that  the  multiple  correlations  be- 
tween reading  and  the  combined  effects  of  the  other  two  is  relatively 
lower  than  the  correlation  of  the  combined  effects  of  reading  and  fun- 
damentals on  reasoning,  and  also,  lower  than  the  correlation  of  the 
combined  effects  of  reading  and  reasoning  on  fundamentals.  All  these 
can  be  taken  as  indications  only,  but  they  lead  to  the  belief  that  a 
knowledge  of  fundamentals  is  the  first  and  most  Important  essential  in 
the  arithmetic  situation  and  the  reading  and  reasoning  are  both  of 
great  importance  in  this  connection. 

Practical  Analysis  of  Pupils'  Papers 

In  the  study  of  the  actual  exercises  the  procedure  was  different.  The 
point  of  view  here  was  a  seeking  for  cause,  while  the  statistical  treat- 
ment was  designed  to  bring  out  the  meaning  of  the  data. 

a.  Tabulation  to  find  types  of  examples  missed 

The  practical  treatment  of  the  results  consisted  of  the  following  steps: 

(1)  a  determination  by  the  course  of  study  of  the  examples  on  the 
test  sheet  that  pupils  completing  each  grade  might  be  expected  to  work; 

(2)  a  tabulation  of  the  number  of  pupils  missing  these  examples;  (3) 
a  conversion  of  these  into  percentages;  (4)  an  analysis  of  examples 
missed  by  as  many  as  ten  per  cent  of  the  pupils,  with  a  view  to  de- 
termining types  of  errors,  difficulties  and  interesting  peculiarities  in- 
cident to  teaching  and  learning  fundamentals;  (5)  a  summary  of  con- 
clusions drawn   from  the  study  outlined  above. 

h.  Analysis  of  examples  missed  by  types  of  errors 

Following  this  plan  a  careful  tabulation  of  errors  counting  numbers 
of  pupils  making  each  type  of  error  rather  than  actual  number  of 
errors  made,  the  information  that  follows  was  obtained.  Since  in  many 
cases  of  unfinished  examples  some  types  of  errors  connote  others,  the 
errors   given   in    all   cases   are   a   minimum   number. 


Diagnostic  Study  in  Arithmetic  7 

In  general  pupils  entering  grade  4  from  grade  3  may  be  expected 
to  work  3  2  examples  on  the  test  paper,  distributed  as  follows:  sub- 
traction, examples  1-17;  multiplication  1-18;  addition  1-16;  division 
1-19.  Pupils  in  grade  3  have  failed  to  fix  examples  of  the  type  13,  14, 
and  17,  in  subtraction,  8,  9,  11,  13,  16  and  18  in  multiplication.  7, 
10,  13,  14,  16  in  addition,  and  1,  2,  7,  8,  11,  14,  15,  17,  19  in  division. 

Several  interesting  facts  were  observed.  In  multiplication  by  zero 
pupils  so  often  say  "one  times"  the  number.  Often  the  minuend  was 
taken  for  the  subtrahend  when  the  latter  was  larger,  as  16  —  9  =  13. 
The  answer  for  an  example  of  equation  form  was  often  reversed,  as 
14  of  128  z=  23.  A  number  times  zero  is  often  given  as  one  times  the 
number.  When  there  is  a  zero  in  the  multiplier  and  a  carried  number 
is  to  be  added  to  the  multiplier  times  it,  pupils  often  put  down  the  zero 
only. 

Types  of  errors  most  generally  made  in  the  beginning  of  grade  4  in 
order  of  importance  as  shown  by  the  tabulation  of  errors  are  as  follows: 

In  subtraction,  errors  are  due  to  ignorance  of  the  more  difficult  sub- 
traction facts,  ignorance  of  the  type  of  exercises  with  two  figures  in 
minuend  and  subtrahend,  incorrect  borrowing  and  incorrect  handling  of 
the   zero   in   subtraction. 

Errors  in  multiplication  are  due  to  insufficient  knowledge  of  the  more 
difficult  multiplication  facts,  incorrect  handling  of  zero  in  multiplier 
and   multiplicand,   and   incorrect   carrying   where   multiplicand    is    larger. 

In  division,  errors  are  due  to  ignorance  of  the  more  difficult  division 
facts,  zero  difficulties  in  division,  trouble  with  remainder,  unfamiliar 
form    (equation)   and  inability  to  handle  exercises  with  large  dividends. 

In  addition,  errors  are  due  to  ignorance  of  the  more  difficult  additive 
facts,  incorrect  carrying,  errors  in  higher  decade  addition,  unfamiliar 
form  and  ignorance  of  type  of  exercise  with  two  figures  or  more  In 
addends. 

For  pupils  entering  grade  5  from  grade  4  it  was  found  that  in  gen- 
eral they  may  be  expected  to  work  41  examples  on  the  test  sheet,  dis- 
tributed as  follows:  1-19  in  subtraction,  1-18  in  multiplication,  1-16, 
22,  28  In  addition,  and  1-23  in  division.  Pupils  in  grade  4  have  failed 
to   fix  examples   of  the  types   14,    17,    19,   in   subtraction;    9,    11,    12,    13, 

16,  18,   26   in  multiplication;    14,   16,   22,   23,  in  addition  and   11,   14,   15, 

17,  19,   23,   28  in  division. 

Several  interesting  points  were  observed.  Pupils  sometimes  put  down 
whole  number  in  the  product  instead  of  carrying  this  number.  A  very 
prevalent  error  is  giving  the  answer  in  long  division  with  zero  omitted 
as  2  9/29  for  20  9/29.  Zero  is  sometimes  put  in  quotient  where  it 
does  not  belong  as  7032  for  732.  Where  the  number  is  carried  and 
there  is  a  zero  in  the  multiplier  pupils  often  put  zero  in  product,  in- 
stead of  the  number  carried.  Remainders  are  often  placed  over  divi- 
dend instead  of  divisor.  In  addition  carried  number  is  sometimes  drawn 
down  as  5  29  for  79.  Often  a  zero  in  multiplier  causes  wrong  placing 
of  products.  Errors  in  the  more  difficult  combinations  sometimes  connote 
errors  in  carrying.  Where  there  are  errors  in  estimating  quotient  these 
would  usually  be  accompanied  by  errors  in  handling  remainders  but 
since  the  example  is  unfinished  it  is  impossible  to  tell.      In  the  same  way 


8  Diagnostic  Study  in  Arithmetic 

errors  in  bridging  nearly  always  connote  errors  in  carrying.  Example 
23  in  division  contains  a  zero  difficulty  but  since  it  was  tried  by  so 
few  the  errors  could  not  be  tabulated.  Therefore  the  errors  tabulated 
in  each  case  are  the  minimum  number. 

Types  of  errors  made  by  pupils  completing  grade  4  and  entering  grade 

5  are: 

In  subtraction,  errors  are  due  to  ignorance  of  the  more  difficult  sub- 
traction combinations,  reversing  process  when  subtrahend  digit  is  larger 
than  corresponding  digit  in  minuend,  those  due  to  zero  difficulties,  bor- 
rowing, unfamiliar  equation  forms,  confused  process  and  careless  errors 
(copying,    etc.) 

In  multiplication,  types  of  errors  made  are  those  due  to  ignorance  of 
more  difficult  multiplicative  facts,  carrying  zero  difficulties,  unfamiliar 
equation  form,  confusion  of  process  (harmful  transfer)  incorrect  placing 
of  partial  products,  and  careless  errors  (such  as  placing  decimal  where 
they  do  not  belong,  copying,  etc.) 

In  addition,  types  of  errors,  are  those  due  to  ignorance  of  the  more 
difficult  addition  combinations,  inability  to  handle  higher  decade  ad- 
dition, zero  difficulties,  carrying,  unfamiliar  equation  form,  confused 
process,   and   careless   errors. 

For  pupils  entering  grade  6  from  grade  5  it  was  found  that  in  gen- 
eral pupils  may  be  expected  to  work  48  examples  distributed  as  follows: 
1-20  in  subtraction,  1-18  in  multiplication,  1-23  in  addition,  and  1-27  in 
division.  Pupils  in  grade  5  have  failed  to  fix  examples  of  the  type  1, 
7,  19,  20  in  subtraction,  8-18,  24,  26  in  multiplication,  20-23  in  addition 
and  17,  19,  23,  27  in  division. 

Several  interesting  facts  were  observed.  Remainders  were  often  put 
over  the  dividend  instead  of  the  divisor.  Errors  in  estimating  quotients 
usually  connote  errors  in  bringing  down  terms  of  dividend.  Subtraction 
in  long  division  is  often  wrong  because  figures  are  not  put  in  the  right 
places   to   be   subtracted. 

Types  of  errors  in  subtraction  made  by  pupils  entering  grade  6  are 
those  due  to  ignorance  of  more  difficult  subtraction  combinations,  incor- 
rect borrowing,  zero  difficulties,  confused  processes,  careless  errors,  failure 
to  reduce  to  lowest  terms,  putting  denominator  under  integer,  unfamiliar 
equation  form,  drawing  down  fractions,  not  multiplying  by  numerator 
or  dividing  by  denominator,  and  subtracting  the  denominator  from  the 
multiplicand. 

Types  of  errors  in  multiplication  are:  more  difficult  combinations,  car- 
rying, zero,  partial  products,  confused  errors  in  computation,  failure  to 
reduce  to  lowest  terms,  error  in  reducing  to  whole  or  mixed  number, 
putting  denominator  under  integer,  equation,  drawing  down  fractions, 
not  trying,  not  multiplying  by  numerator  or  dividing  by  denominator, 
and    subtracting    denominator    from    multiplicand. 

Types  of  errors  in  addition  are  those  due  to  ignorance  of  more  difficult 
addition  combinations,  incorrect  carrying,  zero  difficulty,  inability  to 
handle  higher  decade  addition,  confused  process,  careless  errors,  failure 
to  reduce  to  lowest  terms,  unfamiliar  equation  form,  failure  to  place 
decimal,    and    multiplying   denominator. 


Diagnostic  Study  in  Arithmetic  » 

Types  of  errors  in  division  are  those  due  to  ignorance  of  more  difficult 
combinations,  inability  to  estimate  quotients,  zero  difficulties,  incorrect 
bringing  down,  confused  process,  careless  errors,  failure  to  reduce  to 
lowest  terms,  unfamiliar  equation  form,  inability  to  handle  remainders, 
incorrect  placing  of  decimal  and  adding  in  numerator. 

For  pupils  entering  grade  7  from  grade  6  it  was  found  that  in  gen- 
eral pupils  may  be  expected  to  work  55  examples  on  the  test,  distributed 
as  follows:  1-25  in  subtraction,  1-29  multiplication,  1-3  6  in  addition, 
and  1-27  in  division.  Pupils  in  grade  6  have  failed  to  fix  examples  of 
the  type  19,  20,  24,  25  in  subtraction,  12-29  in  multiplication,  16,  20-36 
in  addition,  and  17,  19,  23,  27  in  division. 

Types  of  errors  in  subtraction  made  by  pupils  entering  grade  7  are 
those  due  to  ignorance  of  the  more  difficult  combinations,  incorrect  bor- 
rowing, zero  difficulties,  drawing  down  fractions,  integer  treated  as  a 
fraction,   not   getting  to   common    denominator  and   confused   process. 

Types  of  errors  in  multiplication  are  those  due  to  insufficient  knowledge 
of  more  difficult  combinations,  incorrect  carrying,  zero  difficulties,  in- 
correct handling  of  partial  pi-oducts,  bringing  down  fractions,  incorrect 
placing  of  decimals,  careless,  confused  process  and  failure  to  reduce  to 
lowest  terms. 

Errors  in  addition  are  those  due  to  ignorance  of  more  difficult  addition 
combinations,  incorrect  carrying,  inability  to  do  higher  decade  addition, 
adding  denominator,  unfamiliar  equation  form,  multiplying  denominator, 
not  getting  common  denominator,  inability  to  handle  denominate  num- 
bers, incorrect  placing  of  decimal  and  not  reducing  to  lowest  terms. 

Errors  in  division  are  those  due  to  inability  to  estimate  quotients,  zero 
difficulties,  incorrect  bringing  down,  unfamiliar  equation  form,  inability 
to  handle  remainders,  incorrect  placing  of  decimal,  adding  numerator, 
careless   and    harmful   transfer. 

For  pupils  entering  grade  8  from  grade  7  it  was  found  that  in  gen- 
eral pupils  may  be  expected  to  work  58  examples  distributed  as  follows: 
1-27  in  subtraction,  1-33  in  multiplication,  1-3  6  in  addition  and  1-30 
in  division.  Pupils  in  grade  7  have  failed  to  fix  the  following  types  of 
examples:  2  5,  27  in  subtraction;  18,  26,  27,  29,  33  in  multiplication 
20,  21,  22,  24,  30,  33,  36  in  addition;  and  19,  23,  27,  28,  30  in  division. 
In   many  cases   fractions   were  not  tried. 

Types  of  errors  in  subtraction  made  by  pupils  entering  grade  8  from 
grade  7  are  those  due  to  incorrect  borrowing,  drawing  down  fractions, 
integers  treated  as  fractions,  not  getting  common  denominator,  harm- 
ful  transfer  and  inability  to  handle   denominate  numbers. 

Types  of  errors  in  multiplication  are  those  due  to  insufficient  knowledge 
of  more  difficult  combinations,  incorrect  carrying,  incorrect  handling  of 
partial  products,  drawing  down  fractions,  incorrect  placing  of  decimal, 
failure  to  reduce  to  lowest  terms,  harmful  transfer,  careless,  failure  to 
multiply  by  numerator  or  divide  by  denominator,  and  multiplying  the 
denominator. 

Types  of  errors  in  addition  are  those  due  to  incorrect  carrying,  in- 
correct borrowing,  inability  to  do  higher  decade  addition,  failure  with 
denominate  numbers,  incorrect  placing  of  decimal,  not  reducing  to  lowest 
terms,   and  not  getting  a  common   denominator. 

Types  of  errors  in  division  are  those  due  to  inability  to  estimate  quo- 


10  Diagnostic  Study  in  Auitiimetic 

tients,  zero  difficulties,  incorrect  bringing  down  terms  of  dividend,  in- 
ability to  handle  remainders,  subtraction  in  long  division,  harmful  trans- 
fer,   wrong   inversion    and   incorrect    placing    of   the   decimal. 

Several  conclusions  may  be  briefly  shown  from  the  above  study 
(a)  there  has  been  insufficient  drill  on  the  more  difficult  combinations 
in  all  four  fundamentals;  (b)  some  hard  things  such  as  handling  the 
zero,  long  division,  carrying  and  borrowing  have  been  inadequately  taught 
and  insufficieht  drill  has  been  furnished;  (c)  drills  given  have  not  always 
fitted  children's  needs. 

Individual  Diagnostic  Tests  to  Fiid  Out  Mental  Habits  That  Slow  Up  Work 
and  Make  It  Inaccurate 

The  third  approach  to  the  problem  is  the  use  of  the  individual  diag- 
nostic tests  to  determine  the  mental  habits  that  slow  up  the  pupil's  work 
and  make  it  inaccurate. 

a.  A  test  admirably  suited  for  individual  testing  is  the  Buswell-John. 
The  pupils'  sheet  contains  exercises  involving  all  the  known  difficulties 
in  the  four  fundam^entals.  The  teachers'  sheet  contains  the  same  exer- 
cises with  a  list  of  the  bad  habits  usually  observed  in  each  of  the  four 
fundamiental  operations. 

b.  As  the  pupils  work  aloud  the  examiner  carefully  observes  and 
questions  him  and  is  able  to  discover  his  bad  habits. 

PLAN  OF  WORK,  RESULTS  AND  INTERPRETATION  OF  RESULTS 

In  this  case  the  supervisor  selected  three  children,  one  slow,  one 
average,  and  one  superior,  from  each  of  the  grades  3-7  in  six  schools 
of  widely  varying  types  in  the  county.  This  made  a  group  of  represen- 
tative grades,  each  grade  comprising  eighteen  children:  six  superior,  six 
average,  and  six  slow  pupils.  A  careful  testing  of  these  ninety  pupils 
gave  information  as  to  the  number,  and  kinds  of  bad  habits  prevalent 
in  each  group  in  each  grade,  and  also  showed  types  of  examples  missed 
by  each  group.  These  were  arranged  in  order  of  frequency  as  shown 
as   follows: 

For  Grade  4 

1.  Addition:  Errors  in  combinations,  counting  added  carried  number 
last,  forgot  to  add  carried  number,  repeated  work  after  partly  done, 
carried  wrong  number,  wrote  number  to  be  carried,  irregular  pro- 
cedure in  column,  used  wrong  fundamental  operation,  and  dropped 
back  one  or  more  tens. 

2.  SuMr action:  Errors  in  combinations,  did  not  allow  for  having  bor- 
rowed, counting,  subtracted  minuend  from  subtrahend,  put  zero  in 
front  of  answer,  (as  follows,  0  6  for  6),  failed  to  borrow  (gave  zero 
as   answer),   added   instead   of   subtracting,    and   ignored   a   digit. 

3.  Multiplication:  Errors  in  combinations,  errors  due  to  zero  in  mul- 
tiplier, errors  in  addition,  errors  in  single  zero  combinations  (zero 
as  multiplier),  omitted  digit  in  multiplier,  error  in  adding  the  car- 
ried number,  carried  a  wrong  number,  forgot  to  carry,  counted  to 
carry,  wrote  rows  of  zeros,  errors  in  position  of  partial  product, 
errors  in  writing  product,  used  wrong  process  (added),  wrote  car- 
ried number,  confused  products  when  multiplier  had  two  or  more 
digits,   and  errors  in  carrying  zero. 


Diagnostic  Study  in  Arithmetic  11 

4.  Division:  Errors  in  division  combinations,  errors  in  subtraction, 
errors  in  multiplication,  found  quotient  by  trial  multiplication, 
omitted  digit  in  dividend,  used  remainder  larger  than  divisor,  omitted 
final  remainder,  omitted  zero  resulted  from  another  digit,  not  re- 
ducing remainder  to  lowest  terms,  neglected  to  use  remainder  w^ithin 
problem,  used  long  division  for  short  division,  and  used  too  large 
a  product. 

Examples    missed    by   pupils,   in    order   of   frequency   are:     (See    Buswrell- 
John  Test  Sheet). 

For  superior  group  of  pupils: 

Addition:    21,    19,    5,    18,    20,    22,    23. 
Subtraction:    17. 

Multiplication:   16,  17,  18,  14,  15. 
Division:    16,   17,  15,  11,   12,  13,   14,  8,   9. 

For  average  group  of  pupils: 
Addition:    19,   20,   21,   22,   8. 

Subtraction:    15,    18,   8,    9,   10,    11,    12,   13,   14,   16. 
Multiplication:    16,   17,   18,   12,   9,   6,   13,   15,   14. 
Division:    16,   17,   10,  12,   14,  15,  13,  8,   9. 

For  -sloMT  pupils: 

Addition:    21,  22,  19,  18,  4,  6,  9,  14,  17,  10,  12,  15,  20,  23. 
Subtraction:    11,   18,   14,   17,   15,   4,   5,   12,   13,   16. 
Multiplication:    13,   14,   15,   16,  17,   11,   12,   9,   18,   5,   2,   10. 
Division:   13,  14,  15,  16,  17,  10,  5,  11,  12,  2,  3,  6,  8,  9. 

For  Grade  5 

1.  Addition:  Errors  in  combinations,  added  carried  number  last,  count- 
ing, forgot  to  add  carried  nuinber,  repeated  M^ork  after  partly  done, 
irregular  procedure  in  column,  carried  wrong  number,  and  wrote  the 
number  to  be  carried. 

2.  Salitraction:  Did  not  allow  for  having  borrov/ed,  errors  in  com- 
binations, counting,  deducted  two  from  minuend  after  borrowing, 
and   subtracted   minuend   from   subtrahend. 

3.  MuJti2)lication:  Errors  in  combinations,  carried  a  wrong  number, 
error  in  adding  carried  number,  errors  due  to  zero  in  multiplier, 
omitted  digit  in  multiplier,  error  in  single  zero  combinations  (zero 
as   multiplier)    and   wrote   carried   number. 

4.  Division:  Errors  in  division  combinations,  found  quotient  by  trial 
multiplication,  omitted  zero  resulting  from  another  digit,  errors  in 
subtraction,  used  remainder  larger  than  divisor,  used  long  divis- 
ion form  for  short  division,  omitted  final  remainder,  failed  to  reduce 
to  lowest  terms  (remainder),  omitted  digit  in  dividend,  and  re- 
peated part  of  multiplication  table. 

Examples  missed  by  pupils  in  order  of  frequency  are: 
For  superior  pupils: 

Addition:    21,    22,    18,    23,    14,    16. 
Multiplication:    14,    17,    22,    13,    15,    16,    18. 
Division:    17,   16,    18,   21,   12,   15,   20. 


12  Diagnostic  Study  in  Arithmetic 

For  average   group  of  pupils: 

Addition:    21,   16,   18,   19,   22,   12,   15,   20,   23. 
Subtraction:    18,   21,   22,   13,   8,   15,   17,   19,   20. 
Multiplication:    13,  15,   17,   18,   19,   20,   22,   12,   16,   21,   9,   14. 
Division:    19,   16,   17,   18,   21,   13,   14,   15,   11,   20,   12,   10. 

For  slow  group  of  pupils: 

Addition:  21,  22,  19,  23,  8,  9,  13,  17,  18,  20. 
Subtraction:  17,  18,  21,  22,  10,  19,  20,  13,  15. 
Multiplication:    17,    19,    21,    22,    12,    13,    15,    16,    20,    6,    18,    1,    2,    3, 

5,   11.    7. 
Division:    12,  15,  16,  17,  18,  19,   20,  21,  8,   11,   13,  14,  10,   5,  2,  6,   9. 

For  Grade  6 

1.  Addition:  Errors  in  combinations,  counting,  added  carried  number 
last,  forgot  to  add  carried  number,  repeated  work  after  partly  done, 
wrote  number  to  be  carried,  irregular  product  in  column,  and  added 
carried  number  irregularly. 

2.  SuMraction:  Errors  in  combination,  did  not  allow  for  having  bor- 
rowed, counting  said  example  backwards,  deducted  2  from  the  min- 
uend after  borrowing,  and  put  zero  in  front  of  answer  as  0  6  for  6. 

'  3.  Multiplication:  Errors  in  combinations,  errors  due  to  zero  in  mul- 
tiplication, counted  to  carry,  error  in  single  zero  combinations  (zero 
as  multiplier),  omit  digit  in  multiplier,  errors  in  addition,  wrote 
rows  of  zeros,  carried  a  wrong  number,  and  wrote  carried  numbers. 

4.  Division:  Errors  in  subtraction,  found  quotient  by  trial  multipli- 
cation, used  long  division  for  short  division,  omitted  zero  resulting 
from  another  digit,  errors  in  combinations,  errors  in  multiplication, 
omitted  digit  in  dividend,  not  reducing  remainder  to  lowest  terms, 
counted  in  subtracting,  used"  remainder  larger  than  divisor,  omitted 
final  remainder,  and  neglected  to  use  remainder  within  problem. 

Examples  missed  by  pupils  in  order  of  frequency  are: 

For  superior  group  of  pupils: 
Subtraction:    17,   18,   20,   26. 
Multiplication:    14,   16,    19,    20,    7,    13,   18. 
Division:    21,   16,   17,   18,   10,   13, 

For  average  group  of  pupils: 

Subtraction:    19,    21,    16,    22,    14,    15,    18. 

Addition:    21,   19,   22,   18,   23,    3,   8,   17,   20. 

Multiplication:    19,   21,   16,    22,   14,   15,   18. 

Division:    20,    21,   15,   16,   17,    12,   13,   18,   10,   11,    13,   19. 

For  slow  group  of  pupils: 

Addition:    21,   19,   23.   8,    17,   4,   6,   10,   18,    20,    22. 
Subtraction:    19,   20,   22,   15,   17,   13,   21,   10,   18. 
Multiplication:    16,   22,  14,  17,  18,   21,  15,   19,   20,   6,   12. 
Division:    16,   17,   18,   20,   21,   8,   11,   12,   13,   19,   1,   9,   10,   14. 


Diagnostic  Study  in  Arithmetic  13 

For  Grade  7 

1.  Addition:  Errors  in  combinations,  added  carried  number  last,  count- 
ing, repeated  work  after  partly  done,  and  wrote  number  to  be 
carried. 

2.  Subtraction:  Errors  in  combinations,  did  not  allow  for  having  bor- 
rowed, and  counting. 

3.  Multiplication :  Errors  in  combinations,  errors  in  single  zero  com- 
binations (zero  as  multiplier),  errors  in  addition,  counted  to  carry, 
error  in  adding  the  carried  number,  wrote  rows  of  zeros,  wrote  car- 
ried number,  errors  due  to  zero  in  multiplier,  forgot  to  carry  and 
omitted  digit  in  multiplier. 

4.  Division:  Found  quotient  by  trial  multiplication,  errors  in  mul- 
tiplication, errors  in  subtraction,  used  long  division  combinations, 
omitted  digit  in  dividend,  omitted  zero  resulting  from  another  digit, 
counted  in  subtracting,  did  not  reduce  remainder  to  lowest  terms, 
used  remainder  larger  than  divisor,  and  used  digit  in  dividend  twice. 

Examples  missed  by  pupils  in  order  of  frequency  are: 
For  superior  group  of  pupils: 
Addition:    21,   19,   22. 
Subtraction:    18. 
Multiplication:    21,   2  2. 
Division:    20,  13,  17,  21,  15,  18. 

For  average  group  of  pupils: 
Addition:    21,  8,  18,  19,  22. 
Subtraction:    18,   19,   22,   14,   15,   17. 
Multiplication:    14,   21,   15,   16,   18,   19,   20. 
Division:    17,   21.   15,   16,    18,   19,   20. 

For  slow  group  of  pupils: 

Addition:    21,   18,    19,    23,    8,    10,    15,    20,    22. 
Subtraction:    18,   21.   22,   15,   16,   17,   19,   20. 
Multiplication:    22,    13,    14,   19,    21,    15,   16,   18,   17. 
Division:    21,   17,   19,   15,   16,   IS,   10,   13,   14,   11,   8,   12,   20. 

It  was  observed  that  slow  pupils  had  far  more  bad  habits  than  average 
pupils,  and  that  average  pupils  had  more  bad  habits  than  superior  pupils. 
Also  superior  pupils  worked  more  exercises  than  average  pupils,  and 
average  pupils  surpassed  slow  pupils  in  this  respect.  Another  significant 
point  was  that  the  schools  varied  among  themselves  in  achievement  and 
number  of  bad  habits  displayed.  In  general  pupils  in  schools  with  small 
teacher  load,  more  highly  trained  teachers,  and  superior  teaching  morale 
showed  relatively  greater  achievement  and  fewer  bad  habits  of  work. 
Pupils  in  one  school,  whose  teachers  are  nearly  all  of  the  older,  more  con- 
servative type,  living  in  the  community,  showed  a  larger  number  of  bad 
habits  of  work.  This  may,  or  may  not  be  significant.  Small  schools  dis- 
played a  relatively  large  number  of  bad  habits  of  work. 

Another  interesting  point  is  that  far  more  habits,  not  listed  on  the 
Buswell-John    test,    were    discovered.      These    are:     (1)    Not   reducing   re- 


14 


DiAGiSiOSTic  Study  in  Arithmetic 


mainder  to  lowest  terms  (d.3  6)  ;  (2)  putting  a  zero  in  front  of  the  answer 
in  subtraction  (s.  25);  (3)  not  placing  remainder  over  divisor  (d.  37); 
and   (4)  writing  borrowed  number   (s.  26). 


The  accompanying  graphs  illustrate  the  comparisons  given  above  as 
to  bad  habits  displayed  and  achievement. 

These  are  significant  in  that  they  shed  light  on  variations  in  instruc- 
tion in  different  types  of  schools  in  the  county  and  on  variations  in 
habits   and   achievement   of   groups  within   the   school. 


Diagnostic  Sti/iiy  in  Asitiimetic 


15 


SUMMARY  OF  DATA  AND  BEARIN«  ON  PROBLEM 

From  the  foregoing  interpretation  of  data  we  may  draw  the  following 
conclusions  in   regard  to  the  problem; 

1.  The  chief  cause  of  retardation  in  arithmetic  is  the  formation  of 
bad  habits  of  work  that  make  the  work  slow  and  inaccurate.  These  habits 
are  due  partly  to  negligence  and  ignorant  procedure  on  the  part  of  the 
teacher,  and  partly  to  habits  formed  independently  by  pupils.  These 
habits  are  usually  not  evident  except  to  the  trained   observer. 

2.  Another  cause  is  the  inadequate  teaching  of  certain  difficult  ma- 
terial and  inadequate  drill  on  newly  learned  material,  especially  the  com- 
binations. 

3.  Another  cause  is  failure  to   classify  pupils  properly. 

4.  A  fourth  cause  is  ignorance  on  the  part  of  the  teacher  of  the 
exact  difficulties  in  her  class,  which  has  prevented  a  close  adjustment  of 
the  instruction  to  the  needs  of  the  pupils. 

USE  OF  DATA  IN  SUGGESTING  REMEDIAL  MEASURES 

Having  completed  the  first  two  steps  in  the  study  (i.e.  (1)  finding  by 
a  study  of  the  data  what  conditions  exist;  and  (2)  investigating  the 
causes  of  such  conditions),  the  third  step  is  to  plan  definite  remedial 
action.  Here  follows  a  few  particular  suggestions  for  the  use  of  the 
information  gained.  First,  and  most  important  of  all,  definite  remedial 
measures  should  be  worked  out  to  counteract  each  bad  habit  discovered 
in  the   pupils.      These   should   be   constantly  revised  and  added  to.      The 


16  DiAGJs'OSTic  Stuoy  i]Sf  Arithmetic 

importance  of  the  co-operation  of  the  child  in  this  work  cannot  be  stressed 
too  heavily.  This  is  a  list  of  remedial  measures  that  have  been  used 
and  found  valuable,  though  it  is  thought  best  that  the  teacher  be  en- 
couraged to  use  her  own  originality  in  devising  remedial  measures. 

Remedial  Suggestions 

1.  Often  let  an  individual  pupil,  usually  a  slow  one,  work  aloud  at 
board,  pupils  and  teacher  noting  good  and  bad  habits.  Discuss  these 
habits  with  pupils,  let  them  see  why  they  hinder  the  work  and  arouse 
a  desire  for  self-improvement.      This  is  very  important. 

2.  Develop  self-control  attitude  in  pupils  and  have  each  pupil  work 
on  his  own  difficulties. 

3.  When  teaching  a  new  process  be  sure  to  point  out  good  and  poor 
methods  of  procedure,  giving  advantages  and  disadvantages.  Give 
careful  explanation  of  new  process. 

4.  Give  drill  only  when  material  is  thoroughly  understood. 

5.  Use  any  device  you  can  think  of  to  arouse  pupils'  interest:  games, 
posters,    competition,    projects,    etc. 

6.  In  reteaching  types  of  examples  imperfectly  understood,  be  sure  to 
point  out  repeatedly  what  causes  difficulty,  as  for  instance  the  prev- 
alent habit  of  leaving  out  a  zero  in  the  quotient. 

7.  Slow  pupils  have  more  bad  habits  of  work  than  average  or  bright 
ones,   so  work  especially  with  these  pupils. 

8.  Co-operative  work  at  the  blackboard  and  checking  helps.  Let  one 
pupil  perform  one  step,  another  one  the  next  and  so  on. 

9.  Present  examples  in  such  order  that  only  one  new  difficulty  is  pre- 
sented at  a   time. 

10.  Pupils'  names  and  combinations  causing  them  difficulty  may  be 
placed  on  the  board  for  reference  and  study.  Encourage  pupils  to 
work  on  these  before  and  after  school  and  at  study  period. 

11.  Each  pupil  might  keep  record  of  his  errors  in  a  little  book. 

12.  To  remedy  irregular  column  procedure  examples  might  be  written 
in  words. 

13.  Using  concrete  numbers  instead  of  abstract  ones  often  helps  when 
a  pupil  has  zero  difficulties.      (For  instance,  marbles,  apples,  etc.) 

14.  Occasionally  going  through  examples  slowly  in  correct  form  with 
whole  class  helps  to  form  correct  habits. 

15.  Commend  good  form  in  work  whenever  found. 

16.  Observe  what  bad  habits  your  pupils  have  and  try  to  work  out 
something  to  combat  each  bad  habit.  Use  any  method  you  can 
think   of. 

17.  The  Scott-Foresman  work  books  are  excellent  help,  both  for  diag- 
nosis and  drill.      Good  drill  material  can  be  made  by  the  teacher. 

18.  Try  to  prevent  careless  errors  by  arousing  pupils'  pride. 

19.  Keep  a  combination  posted  until  it  is  mastered  (can  be  given  with- 
out   hesitation). 

20.  Remember  that  the  time  to  forestall  bad  habits  is  when  the  material 
is  first  presented.  The  pupils  will  make  up  for  themselves  poor 
and  time-wasting  methods  that  we  must  discover  and  get  rid  of. 
Demonstrate  the  advantage  of  a  better  method. 


Diagnostic  Study  in  Arithmetic  17 

21.  Gaining  pupils'  co-operation  will  make  it  easier  to  discover  bad  habits 
and  easier  to  get  rid  of  them. 

NOTE:  Ask  the  teacher  to  write  down  any  method  she  finds  effective. 
At  the  close  of  the  school  term  ask  her  to  report  what  specific 
things  she  did  for  the  bad  habits  found  during  the  term.  Some  most 
excellent  suggestions  for  remedial  work  are  given  by  G.  T.  Buswell 
in  his  "Diagnostic  Studies  in  Arithmetic" — University  of  Chicago. 

SUGGESTIONS  FOR  FURTHEK  STUDY  AND  EXPERIMENTATION 

At  the  beginning  of  the  year  teachers  should  be  furnished  with  list 
of  errors  similar  to  ones  given  in  first  part  of  this  paper — arranged  in 
order  of  prevalence.  The  teacher  should  use  this  list  as  a  guide  in  her 
review  work  at  the  beginning  of  the  term  since  it  shows  what  the  pupils 
have  failed  to  get  in  the  preceding  grade.  These  same  lists  will  indi- 
cate (from  last  year's  failures)  what  should  be  especially  stressed  this 
year. 

All  teachers  of  grade  3-8  inclusive  should  lay  special  stress  on  the 
more  difficult  combinations  in  all  four  fundamentals.  Special  practice 
cards  would  be  of  service  here  if  they  were  supplemented  by  extra  drill 
on  combinations  missed  by  a  particular  class. 

Rem.edial  drills  should  be  modeled  on  the  types  of  exercises  that  the 
study  showed  that  pupils  had  failed  to  fix.  Diagnostic  drills  should  be 
made  accordingly. 

Teachers  should  demand  absolute  accuracy,  in  view  of  the  large  num- 
ber of  careless  errors  found. 

Also  the  teacher  should  always  be  on  the  lookout  for  individual  dif- 
ficulties and  correct  these  at  once.  It  is  felt  that  what  will  bring  about 
a  great  deal  of  benefit  in  this  particular  situation  is  the  finding  of  the 
needs  of  the  children  and  fitting  instruction  and  drills  closely  to  them. 

In  this  diagnostic  study,  the  teachers  then  undertook,  with  the  aid 
of  the  supervisor,  to  put  the  remedial  suggestions  into  effect.  Frequent 
checks  were  made  by  the  teachers,  and  at  the  close  of  the  year  another 
form  of  the  survey  test  (Woody  Fundamentals)  was  used  to  measure 
progress.  It  was  found  that  the  average  retardation  in  this  subject  was 
5.4  month  instead  of  8.0  as  previously.  It  was  felt  by  supervisor  and 
teachers  that  the  right  method  was  being  employed  and  should  be  con- 
tinued the  next  year  and  carried  beyond  the  field  of  integers.  The  pupils' 
difiiculties  in  fractions  indicated  that  a  similar  diagnosis  of  fractional 
diflaculties  would  be  most  profitable.  The  next  step  after  that  would 
logically  be  an  analysis  of  problem  difficulties  and  study  of  the  technique 
of  problem  solving.  At  present  a  diagnostic  test  in  fractions  is  not  on 
the  market,  but  a  satisfactory  one  could  be  made  by  the  teacher. 

TESTS  FOR  SURVEY  AND  DIAGNOSTIC  PURPOSES 

A  word  here  would  not  be  out  of  place  concerning  the  advantages  and 
disadvantages  of  the  tests  used  in  the  study.  The  survey  test  used  had 
the  obvious  advantage  of  being  a  practical  testing  medium  for  a  large 
number  of  pupils  and  easily  scored.  The  resulting  scores  were  suited 
to  a  rather  extensive  statistical  treatment.  Progress  was  easily  measured 
with  this  test.  Its  limitation  is  that  it  gives  a  vague  picture  of  the 
pupil's  difficulties.  The  Buswell-John  diagnostic  test,  on  the  other  hand, 
is  impractical  for  a  large  number  of  pupils,   since  it  must  be  given  in- 


18  Diagnostic  Study  in  Aritiimetk 

dividually.  Its  advantage  lies  in  the  fact  that  the  pupil  works  aloud 
and  the  examiner  observes  the  way  his  mind  works  when  encountering 
the  exercise.  One  test  supplements  the  other.  Briefly,  the  survey  test 
shows  what  the  pupils  can  do;   the  diagnostic  test  shows  how  he  does  it. 

BIBLIOGRAPHY 

Corrective   Arithmetic — Osborne. 

Diagnostic  Studies  in  Arithmetic — Buswell. 

Teaching  Number  Fundamentals — Hillegas. 

Educational   Statistics — Odell. 

An   Arithmetic   for   Teachers — Rountree   and   Taylor. 

Teaching  Arithmetic  in  the  Primary  Grades — Morton. 

Teaching  Arithmetic  in  the   Intermediate   Grades — Morton. 

Some  Types  of  Difficulties  in  the  Supervision  of  Arithmetic — Ethel  Blair 
Garrett- — Peabody   Contribution. 

Beginnings  in  Educational   Measurement — Lincoln. 

New  Methods  in  Arithmetic — Thorndike. 

Modern  Arithmetic  Methods   and   Problems — Lindquist. 

Elementary  Arithmetic — Myers. 

Applied  Arithmetic — Lennes  and   Jenkens. 

Special  Methods  in  Arithmetic — McMurry. 

The  Teaching  of  Modern  Day  Arithmetic — McNair. 

Summary  of  Educational  Investigations  Relating  to  Arithmetic — Mon- 
ograph— Judd  and  Buswell. 

Third  Yearbook  of  the  Department  of  Superintendence  of  the  N.  E.  A. 

Prevention  and   Correction  of  Errors  in  Arithmetic — Myers. 

Types   of  Elementary  Teaching  and  Learning — Parker. 

Journal  of  Education  Psychology  XVI — Knight. 

Educational  Research  Bulletin  IV,  April  15,  1925 — Ohio  State  Uni- 
versity. 

The  Psychology  of  Arithmetic — Thorndike. 


Binder 
Gaylord  Bros.  Inc. 

Makers 
Syracuse,  N.  Y. 

PAT.  JAN  21,  1908 


00034036365 

FOR  USE  ONLY  IN 
THE  NORTH  CAROLINA  COLLECTIO] 


Form  No.  A-368,  Rev.  8/95 


